The Strong Nullstellensatz for Certain Normed Algebras

Consider the polynomial ring in any finite number of variables over the complex numbers, endowed with the $\ell_1$-norm on the system of coefficients. Its completion is the Banach algebra of power series that converge absolutely on the closed polydisc. Whereas the strong Hilbert Nullstellensatz does not hold for Banach algebras in general, we show that it holds for ideals in the polynomial ring that are closed for the indicated norm. Thus the corresponding statement holds at least partially for the associated Banach algebra. We also describe the closure of an ideal in small cases.